Finite Group Databasebeta

← C4×#89418 #13274995

Subgroups — C4×#89418 #13274995

Order 1792.

2110 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic14C211yesmaximal, characteristic
2C2×C4×Dic7⋊C4C211yesmaximal, characteristic
3C2×C42×Dic7C211yesmaximal, characteristic
4C2×#89418C212yesmaximal
5C4×C7×#95C211yesmaximal, characteristic
6C4×C7⋊#95C211yesmaximal, characteristic
7C4×C7⋊#95C211yesmaximal, characteristic
8C4×C7⋊#95C211yesmaximal, characteristic
9#11727161C212yesmaximal
10#11727160C212yesmaximal
11Dic7⋊#95C212yesmaximal
12C23×Dic14C2211yescharacteristic
13C22×C42×C7C2211yescharacteristic
14C22×C42×C7C2211yescharacteristic
15C22×C42×C7C2211yescharacteristic
16C22×C4⋊Dic7C2211yescharacteristic
17C22×Dic7⋊C4C2211yescharacteristic
18C22×Dic7⋊C4C2211yescharacteristic
19C22×Dic7⋊C4C2212yes—
20C22×C4×Dic7C2212yes—
21C22×C4×Dic7C2211yescharacteristic
22C22×C4×Dic7C2211yescharacteristic
23C22×C4×Dic7C2212yes—
24C22×C4×Dic7C2211yescharacteristic
25C22×C4×Dic7C2211yescharacteristic
26C2×C4×Dic14—24no—
27C2×C4×Dic14—24no—
28C2×C4×Dic14C418yescomplemented
29C2×C7×#95C2212yes—
30C2×C7×#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37C2×C7⋊#95C2212yes—
38C4×Dic7⋊C4—24no—
39C42×Dic7—24no—
40#89418C4116yesdirect product factor
41C4×#391—71nomaximal, Sylow
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2312yes—
44C23×C4×C7C2311yescharacteristic
45C23×C4×C7C2312yes—
46C23×C4×C7C2311yescharacteristic
47C23×Dic7C2311yescharacteristic
48C23×Dic7C2312yes—
49C23×Dic7C2311yescharacteristic
50C23×Dic7C2311yescharacteristic
51C23×Dic7C2312yes—
52C23×Dic7C2311yescharacteristic
53C22×Dic14—22no—
54C22×Dic14—22no—
55C22×Dic14—24no—
56C22×Dic14C2×C414yes—
57C22×Dic14C2×C418yes—
58C2×C42×C7—22no—
59C2×C42×C7—22no—
60C2×C42×C7—22no—
61C2×C42×C7—22no—
62C2×C42×C7—22no—
63C2×C42×C7—22no—
64C2×C42×C7—22no—
65C2×C42×C7C2×C414yes—
66C2×C42×C7Q814yescomplemented
67C2×C4⋊Dic7—22no—
68C2×C4⋊Dic7—22no—
69C2×C4⋊Dic7C2×C414yes—
70C2×Dic7⋊C4—22no—
71C2×Dic7⋊C4—22no—
72C2×Dic7⋊C4—22no—
73C2×Dic7⋊C4—22no—
74C2×Dic7⋊C4—24no—
75C2×Dic7⋊C4—24no—
76C2×Dic7⋊C4C2×C414yes—
77C2×Dic7⋊C4C2×C414yes—
78C2×Dic7⋊C4C2×C418yes—
79C2×C4×Dic7—22no—
80C2×C4×Dic7—22no—
81C2×C4×Dic7—22no—
82C2×C4×Dic7—24no—
83C2×C4×Dic7—22no—
84C2×C4×Dic7—24no—
85C2×C4×Dic7—24no—
86C2×C4×Dic7—24no—
87C2×C4×Dic7—24no—
88C2×C4×Dic7—22no—
89C2×C4×Dic7—22no—
90C2×C4×Dic7—22no—
91C2×C4×Dic7C2×C414yes—
92C2×C4×Dic7C2×C418yes—
93C2×C4×Dic7C2×C414yes—
94C2×C4×Dic7D414yes—
95C2×C4×Dic7D414yes—
96C2×C4×Dic7Q814yescomplemented
97C4×Dic14—232no—
98C7×#95—24no—
99C7×#95C2×C418yes—
100C7⋊#95—24no—
101C7⋊#95—24no—
102C7⋊#95—24no—
103C7⋊#95—24no—
104C7⋊#95C2×C418yes—
105C7⋊#95C2×C418yes—
106C7⋊#95C2×C418yes—
107C22×C4×Q8—71no—
108C2×#391—72no—
109C2×C4×C4⋊C4—71no—
110C2×C43—71no—
111#1436—72no—
112#1436—72no—
113C4⋊#95—72no—
114C4×#95—71no—
115C4×#95—71no—
116C4×#95—71no—
117C4×#95D711yescharacteristic
118C24×C7C2411yescharacteristic
119C22×C4×C7—22no—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—21no—
128C22×C4×C7—21no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7C22×C414yes—
137C22×C4×C7C22×C412yes—
138C22×C4×C7C22×C414yes—
139C22×C4×C7C22×C412yes—
140C22×C4×C7C22×C414yes—
141C22×C4×C7C22×C412yes—
142C22×C4×C7C2×D412yes—
143C22×C4×C7C2×D412yes—
144C22×C4×C7C2×Q814yes—
145C22×C4×C7C2×Q812yes—
146C22×C4×C7C2×Q812yes—
147C22×C4×C7#4114yes—
148C22×C4×C7#4112yes—
149C22×C4×C7#4112yes—
150C22×Dic7—22no—
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—22no—
168C22×Dic7—22no—
169C22×Dic7C22×C414yes—
170C22×Dic7C22×C412yes—
171C22×Dic7C22×C414yes—
172C22×Dic7C22×C412yes—
173C22×Dic7C22×C412yes—
174C22×Dic7C22×C414yes—
175C22×Dic7C22×C412yes—
176C22×Dic7C22×C414yes—
177C22×Dic7C2×D412yes—
178C22×Dic7C2×D412yes—
179C22×Dic7C2×D412yes—
180C22×Dic7C2×D412yes—
181C22×Dic7C2×Q814yes—
182C22×Dic7#4114yes—
183C22×Dic7#4114yes—
184C2×Dic14—28no—
185C2×Dic14—28no—
186C2×Dic14—216no—
187C2×Dic14—216no—
188C2×Dic14C42116yescomplemented
189C42×C7—28no—
190C42×C7—28no—
191C42×C7—44no—
192C4⋊Dic7—28no—
193Dic7⋊C4—28no—
194Dic7⋊C4—28no—
195Dic7⋊C4—216no—
196C4×Dic7—216no—
197C4×Dic7—28no—
198C4×Dic7—28no—
199C4×Dic7—216no—
200C4×Dic7—44no—
201C4×Dic7C22⋊C4116yes—
202C23×Q8—71no—
203C2×C4×Q8—78no—
204C2×C4×Q8—144no—
205C2×C4×Q8—144no—
206C22×C4⋊C4—71no—
207C22×C4⋊C4—71no—
208C22×C4⋊C4—71no—
209C22×C4⋊C4—72no—
210C22×C42—71no—
211C22×C42—72no—
212C22×C42—71no—
213C22×C42—71no—
214C22×C42—72no—
215C22×C42—71no—
216C22×C42C2×D711yescharacteristic
217C22×C42C2×D711yescharacteristic
218C22×C42C2×D711yescharacteristic
219#391—716no—
220C4×C4⋊C4—144no—
221C2×#95—72no—
222C2×#95—72no—
223C2×#95—72no—
224C2×#95—72no—
225C2×#95—72no—
226C2×#95—72no—
227C2×#95—72no—
228C2×#95C2×D712yes—
229C2×#95C2×D712yes—
230C43—144no—
231C23×C7C23×C411yescharacteristic
232C23×C7C23×C412yes—
233C23×C7C22×D411yescharacteristic
234C23×C7C22×D411yescharacteristic
235C23×C7C22×Q811yescharacteristic
236C23×C7C22×Q811yescharacteristic
237C23×C7C2×#4111yescharacteristic
238C23×C7C2×#4112yes—
239C23×C7C2×#4111yescharacteristic
240C23×C7C2×#4112yes—
241C23×C7C2×#4112yes—
242C2×C4×C7—22no—
243C2×C4×C7—22no—
244C2×C4×C7—24no—
245C2×C4×C7—24no—
246C2×C4×C7—24no—
247C2×C4×C7—24no—
248C2×C4×C7—24no—
249C2×C4×C7—24no—
250C2×C4×C7—24no—
251C2×C4×C7—24no—
252C2×C4×C7—22no—
253C2×C4×C7—24no—
254C2×C4×C7—22no—
255C2×C4×C7—24no—
256C2×C4×C7—22no—
257C2×C4×C7—24no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7C2×C4214yes—
265C2×C4×C7C2×C22⋊C414yes—
266C2×C4×C7C4×Q814yes—
267C2×C4×C7C4×Q814yes—
268C2×C4×C7C4×Q818yescomplemented
269C2×C4×C7C22⋊Q814yes—
270C2×C4×C7C22⋊Q814yes—
271C2×C4×C7#12414yes—
272C2×C4×C7C4⋊Q814yes—
273C2×Dic7—22no—
274C2×Dic7—22no—
275C2×Dic7—24no—
276C2×Dic7—24no—
277C2×Dic7—24no—
278C2×Dic7—22no—
279C2×Dic7—24no—
280C2×Dic7—22no—
281C2×Dic7—24no—
282C2×Dic7—24no—
283C2×Dic7—24no—
284C2×Dic7—24no—
285C2×Dic7—24no—
286C2×Dic7—24no—
287C2×Dic7—22no—
288C2×Dic7—24no—
289C2×Dic7—22no—
290C2×Dic7—24no—
291C2×Dic7—22no—
292C2×Dic7—24no—
293C2×Dic7—42no—
294C2×Dic7—42no—
295C2×Dic7—42no—
296C2×Dic7—42no—
297C2×Dic7—42no—
298C2×Dic7—42no—
299C2×Dic7C2×C4214yes—
300C2×Dic7C2×C4214yes—
301C2×Dic7C2×C22⋊C414yes—
302C2×Dic7C2×C22⋊C414yes—
303C2×Dic7#11714yes—
304C2×Dic7C4×D418yes—
305C2×Dic7C4×D418yes—
306C2×Dic7C4×Q818yescomplemented
307Dic14—232no—
308C22×Q8—74no—
309C22×Q8—78no—
310C22×Q8—142no—
311C22×Q8—142no—
312C22×Q8—144no—
313C23×C4—71no—
314C23×C4—71no—
315C23×C4—71no—
316C23×C4—72no—
317C23×C4—71no—
318C23×C4—72no—
319C23×C4C22×D711yescharacteristic
320C23×C4C22×D712yes—
321C23×C4C22×D712yes—
322C23×C4C22×D711yescharacteristic
323C23×C4C22×D711yescharacteristic
324C4×Q8—1432no—
325C2×C4⋊C4—74no—
326C2×C4⋊C4—74no—
327C2×C4⋊C4—74no—
328C2×C4⋊C4—78no—
329C2×C4⋊C4—142no—
330C2×C4⋊C4—142no—
331C2×C4⋊C4—142no—
332C2×C4⋊C4—142no—
333C2×C4⋊C4—142no—
334C2×C4⋊C4—142no—
335C2×C4⋊C4—144no—
336C2×C4⋊C4—144no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—74no—
345C2×C42—74no—
346C2×C42—74no—
347C2×C42—74no—
348C2×C42—78no—
349C2×C42—74no—
350C2×C42—144no—
351C2×C42—144no—
352C2×C42—144no—
353C2×C42—144no—
354C2×C42—144no—
355C2×C42—142no—
356C2×C42—142no—
357C2×C42—142no—
358C2×C42—142no—
359C2×C42—142no—
360C2×C42—142no—
361C2×C42—142no—
362C2×C42Dic1414yescomplemented
363C2×C42C4×D714yes—
364#95—24no—
365#95—78no—
366#95—78no—
367#95—78no—
368#95—144no—
369#95—144no—
370#95—144no—
371#95—144no—
372#95C4×D718yes—
373C22×C7C22×C4211yescharacteristic
374C22×C7C22×C22⋊C411yescharacteristic
375C22×C7C2×#11711yescharacteristic
376C22×C7C2×C4×D412yes—
377C22×C7C2×C4×D412yes—
378C22×C7C2×C4×Q812yes—
379C22×C7C2×C4×Q811yescharacteristic
380C22×C7C2×C4×Q811yescharacteristic
381C22×C7C2×C4×Q812yes—
382C22×C7C4×#4111yescharacteristic
383C22×C7C4×#4112yes—
384C22×C7C4×#4111yescharacteristic
385C22×C7C4×#4112yes—
386C22×C7C2×C22⋊Q811yescharacteristic
387C22×C7C2×C22⋊Q811yescharacteristic
388C22×C7#52512yes—
389C22×C7C2×#12411yescharacteristic
390C22×C7#52912yes—
391C22×C7#52912yes—
392C22×C7C2×C4⋊Q811yescharacteristic
393C22×C7#53212yes—
394C22×C7#53312yes—
395C22×C7#53312yes—
396C4×C7—24no—
397C4×C7—28no—
398C4×C7—42no—
399C4×C7—44no—
400C4×C7#39118yescomplemented
401Dic7—24no—
402Dic7—24no—
403Dic7—28no—
404Dic7—42no—
405Dic7—44no—
406Dic7C4×C22⋊C418yes—
407C24C23×D711yescharacteristic
408C2×Q8—716no—
409C2×Q8—148no—
410C2×Q8—148no—
411C2×Q8—1416no—
412C2×Q8—1416no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—72no—
431C22×C4—74no—
432C22×C4—72no—
433C22×C4—72no—
434C22×C4—74no—
435C22×C4—74no—
436C22×C4—74no—
437C22×C4—72no—
438C22×C4—72no—
439C22×C4—72no—
440C22×C4—72no—
441C22×C4—74no—
442C22×C4—74no—
443C22×C4—74no—
444C22×C4—72no—
445C22×C4—142no—
446C22×C4—142no—
447C22×C4—142no—
448C22×C4—142no—
449C22×C4—141no—
450C22×C4—141no—
451C22×C4—142no—
452C22×C4—142no—
453C22×C4—142no—
454C22×C4—142no—
455C22×C4—142no—
456C22×C4—142no—
457C22×C4—142no—
458C22×C4—141no—
459C22×C4—141no—
460C22×C4—142no—
461C22×C4—142no—
462C22×C4—142no—
463C22×C4—142no—
464C22×C4C2×Dic1414yes—
465C22×C4C2×Dic1412yes—
466C22×C4C2×C4×D714yes—
467C22×C4C2×C4×D712yes—
468C22×C4C2×C4×D714yes—
469C22×C4C2×C4×D714yes—
470C22×C4C2×C4×D712yes—
471C22×C4C2×C4×D712yes—
472C22×C4#115714yes—
473C22×C4#115712yes—
474C22×C4D4×D712yes—
475C22×C4D4×D712yes—
476C22×C4#115912yes—
477C22×C4Q8×D712yes—
478C4⋊C4—1416no—
479C4⋊C4—148no—
480C4⋊C4—148no—
481C4⋊C4—148no—
482C42—28no—
483C42—28no—
484C42—44no—
485C42—716no—
486C42—1416no—
487C42—1416no—
488C42—148no—
489C42—148no—
490C42—284no—
491C2×C7C2×C4×C22⋊C411yescharacteristic
492C2×C7C42×Q811yescharacteristic
493C2×C7C42×Q811yescharacteristic
494C2×C7C2×#39111yescharacteristic
495C2×C7#229511yescharacteristic
496C2×C7#229511yescharacteristic
497C2×C7#229711yescharacteristic
498C2×C7C4×C22⋊Q812yes—
499C2×C7C4×C22⋊Q812yes—
500C2×C7C4×#12412yes—
501C2×C7C4×C4⋊Q812yes—
502C23C22×Dic1411yescharacteristic
503C23C22×C4×D711yescharacteristic
504C23C22×C4×D712yes—
505C23C2×#115711yescharacteristic
506C23C2×#115712yes—
507C23C2×D4×D711yescharacteristic
508C23C2×D4×D711yescharacteristic
509C23C2×#115911yescharacteristic
510C23C2×Q8×D711yescharacteristic
511C23D7×#4112yes—
512C23D7×#4112yes—
513Q8—1432no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—74no—
537C2×C4—74no—
538C2×C4—74no—
539C2×C4—78no—
540C2×C4—78no—
541C2×C4—74no—
542C2×C4—74no—
543C2×C4—78no—
544C2×C4—144no—
545C2×C4—142no—
546C2×C4—142no—
547C2×C4—144no—
548C2×C4—142no—
549C2×C4—142no—
550C2×C4—144no—
551C2×C4—144no—
552C2×C4—144no—
553C2×C4—144no—
554C2×C4—144no—
555C2×C4—144no—
556C2×C4—144no—
557C2×C4—144no—
558C2×C4—142no—
559C2×C4—144no—
560C2×C4—142no—
561C2×C4—142no—
562C2×C4—144no—
563C2×C4—144no—
564C2×C4—282no—
565C2×C4—282no—
566C2×C4—282no—
567C2×C4—282no—
568C2×C4—282no—
569C2×C4—282no—
570C2×C4C4×Dic1414yes—
571C2×C4C4×Dic1418yescomplemented
572C2×C4C42×D714yes—
573C2×C4#644114yes—
574C2×C4D7×C22⋊C414yes—
575C2×C4#644814yes—
576C2×C4#645014yes—
577C2×C4#645114yes—
578C2×C4#645914yes—
579C7C4×#39111yescharacteristic, complemented, Sylow
580C22C2×C4×Dic1411yescharacteristic
581C22C2×C4×Dic1412yes—
582C22C2×C42×D711yescharacteristic
583C22C4×#115712yes—
584C22C4×#115711yescharacteristic
585C22C2×#644111yescharacteristic
586C22C2×D7×C22⋊C411yescharacteristic
587C22C2×#644811yescharacteristic
588C22C2×#645011yescharacteristic
589C22C2×#645111yescharacteristic
590C22C2×#645911yescharacteristic
591C22#9020812yes—
592C22D7×#11711yescharacteristic
593C22#9021311yescharacteristic
594C22#9021912yes—
595C22C4×#115912yes—
596C22#9022912yes—
597C22C4×D4×D712yes—
598C22C4×D4×D712yes—
599C22#9023812yes—
600C22#9023912yes—
601C22C4×Q8×D712yes—
602C22#9026912yes—
603C4—24no—
604C4—28no—
605C4—42no—
606C4—44no—
607C4—78no—
608C4—148no—
609C4—144no—
610C4—144no—
611C4—282no—
612C4—284no—
613C4#8941818yesdirect product factor
614C2C42×Dic1411yescharacteristic
615C2C2×#8941811yescharacteristic
616C2#1173082911yescharacteristic
617C2C4×#644112yes—
618C2C4×#644812yes—
619C2C4×D7×C22⋊C411yescharacteristic
620C2#1173100811yescharacteristic
621C2C4×#645011yescharacteristic
622C2#1173106411yescharacteristic
623C2C4×#645112yes—
624C2C4×#645912yes—
subgroupG/H
centerC23×C4C22×D7
commutatorC22×C7C22×C42
FrattiniC24C23×D7
FittingC4×C7×#95C2
socle, prime radical, cosocleC24×C7C24
hypercenterC4×#95D7
hypocenterC7C4×#391
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23×C4 ◁ C4×#95

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
28. C2×C4×Dic14C4semidirect
40. #89418C4direct (N × H)
66. C2×C42×C7Q8semidirect
96. C2×C4×Dic7Q8semidirect
188. C2×Dic14C42semidirect
268. C2×C4×C7C4×Q8semidirect
306. C2×Dic7C4×Q8semidirect
362. C2×C42Dic14semidirect
400. C4×C7#391semidirect
571. C2×C4C4×Dic14semidirect
579. C7C4×#391semidirect
613. C4#89418direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C4×Dic142
C2×C4×Dic7⋊C42
C2×C42×Dic72
C2×#894182
C4×C7×#952
C4×C7⋊#952
#117271612
#117271602
Dic7⋊#952
C4×#3917